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June  2016, 36(6): 3375-3416. doi: 10.3934/dcds.2016.36.3375

Quasisymmetric geometry of the Cantor circles as the Julia sets of rational maps

Weiyuan Qiu 1, , Fei Yang 2, and  Yongcheng Yin 3,

1. 

School of Mathematical Sciences, Fudan University, Shanghai 200433, China
2. 

Department of Mathematics, Nanjing University, Nanjing 210093, China
3. 

Department of Mathematics, Zhejiang University, Hangzhou 310027, China

Received  April 2015 Revised  October 2015 Published  December 2015

We give three families of parabolic rational maps and show that every Cantor set of circles as the Julia set of a non-hyperbolic rational map must be quasisymmetrically equivalent to the Julia set of one map in these families for suitable parameters. Combining a result obtained before, we give a complete classification of the Cantor circles Julia sets in the sense of quasisymmetric equivalence. Moreover, we study the regularity of the components of the Cantor circles Julia sets and establish a sufficient and necessary condition when a component of a Cantor circles Julia set is a quasicircle.
Keywords: quasisymmetrically equivalent., Cantor circles, Julia sets.
Mathematics Subject Classification: Primary: 37F45; Secondary: 37F20, 37F1.
Citation: Weiyuan Qiu, Fei Yang, Yongcheng Yin. Quasisymmetric geometry of the Cantor circles as the Julia sets of rational maps. Discrete & Continuous Dynamical Systems, 2016, 36 (6) : 3375-3416. doi: 10.3934/dcds.2016.36.3375
References:
[1]

A. F. Beardon, Iteration of Rational Functions, Graduate Texts in Mathematics, 132, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-4422-6.  Google Scholar

[2]

M. Bonk, Uniformization of Sierpiński carpets in the plane, Invent. Math., 186 (2011), 559-665. doi: 10.1007/s00222-011-0325-8.  Google Scholar

[3]

M. Bonk, M. Lyubich and S. Merenkov, Quasisymmetries of Sierpiński carpet Julia sets, preprint,, , ().   Google Scholar

[4]

M. Bourdon, Immeubles hyperboliques, dimension conforme et rigidité de Mostow, (French) [Hyperbolic buildings, conformal dimension and Mostow rigidity], Geom. Funct. Anal., 7 (1997), 245-268. doi: 10.1007/PL00001619.  Google Scholar

[5]

M. Bourdon and H. Pajot, Quasi-conformal geometry and hyperbolic geometry, in Rigidity in Dynamics and Geometry, Springer, Berlin, 2002, 1-17.  Google Scholar

[6]

G. Cui, Dynamics of rational maps, topology, deformation and bifurcation, Preprint, May, 2002 (early version: Geometrically finite rational maps with given combinatorics, 1997). Google Scholar

[7]

R. L. Devaney, D. Look and D. Uminsky, The escape trichotomy for singularly perturbed rational maps, Indiana Univ. Math. J., 54 (2005), 1621-1634. doi: 10.1512/iumj.2005.54.2615.  Google Scholar

[8]

A. Douady and J. H. Hubbard, On the dynamics of polynomial-like mappings, Ann. Sci. Éc Norm. Sup., 18 (1985), 287-343.  Google Scholar

[9]

M. Gromov, Hyperbolic groups, in Essays in Group Theory, Math. Sci. Res. Inst. Publ., 8, Springer, New York, 1987, 75-263. doi: 10.1007/978-1-4613-9586-7_3.  Google Scholar

[10]

P. Haïssinsky, Géométrie quasiconforme, analyse au bord des espaces métriques hyperboliques et rigidités, Astérisque, 326 (2009), 321-362.  Google Scholar

[11]

P. Haïssinsky and K. Pilgrim, Quasisymmetrically inequivalent hyperbolic Julia sets, Rev. Mat. Iberoam., 28 (2012), 1025-1034. doi: 10.4171/RMI/701.  Google Scholar

[12]

J. Heinonen, Lectures on Analysis on Metric Spaces, Universitext, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4613-0131-8.  Google Scholar

[13]

M. Kapovich and B. Kleiner, Hyperbolic groups with low-dimensional boundary, Ann. Sci. Éc Norm. Sup., 33 (2000), 647-669. doi: 10.1016/S0012-9593(00)01049-1.  Google Scholar

[14]

B. Kleiner, The asymptotic geometry of negatively curved spaces: Uniformization, geometrization and rigidity, in International Congress of Mathematicians, II, Eur. Math. Soc., Zürich, 2006, 743-768.  Google Scholar

[15]

O. Lehto and K. I. Virtanen, Quasiconformal Mappings in the Plane, Springer-Verlag, Berlin, Heidelberg, New York, 1973.  Google Scholar

[16]

C. McMullen, Automorphisms of rational maps, in Holomorphic Functions and Moduli I, Math. Sci. Res. Inst. Publ., 10, Springer, New York, 1988, 31-60. doi: 10.1007/978-1-4613-9602-4_3.  Google Scholar

[17]

J. Milnor, Dynamics in One Complex Variable: Third Edition, Annals of Mathematics Studies, 160, Princeton Univ. Press, Princeton, NJ, 2006.  Google Scholar

[18]

K. Pilgrim and L. Tan, Rational maps with disconnected Julia sets, Astérisque, 261 (2000), 349-383.  Google Scholar

[19]

W. Qiu, X. Wang and Y. Yin, Dynamics of McMullen maps, Adv. Math., 229 (2012), 2525-2577. doi: 10.1016/j.aim.2011.12.026.  Google Scholar

[20]

W. Qiu, F. Yang and Y. Yin, Rational maps whose Julia sets are Cantor circles, Ergod. Th. & Dynam. Sys., 35 (2015), 499-529. doi: 10.1017/etds.2013.53.  Google Scholar

[21]

N. Steinmetz, On the dynamics of the McMullen family $R(z)=z^m+\lambda/z^l$, Conform. Geom. Dyn., 10 (2006), 159-183. doi: 10.1090/S1088-4173-06-00149-4.  Google Scholar

[22]

L. Tan and Y. Yin, Local connectivity of the Julia sets for geometrically finite rational maps, Sci. China Ser. A, 39 (1996), 39-47.  Google Scholar

show all references

References:
[1]

A. F. Beardon, Iteration of Rational Functions, Graduate Texts in Mathematics, 132, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-4422-6.  Google Scholar

[2]

M. Bonk, Uniformization of Sierpiński carpets in the plane, Invent. Math., 186 (2011), 559-665. doi: 10.1007/s00222-011-0325-8.  Google Scholar

[3]

M. Bonk, M. Lyubich and S. Merenkov, Quasisymmetries of Sierpiński carpet Julia sets, preprint,, , ().   Google Scholar

[4]

M. Bourdon, Immeubles hyperboliques, dimension conforme et rigidité de Mostow, (French) [Hyperbolic buildings, conformal dimension and Mostow rigidity], Geom. Funct. Anal., 7 (1997), 245-268. doi: 10.1007/PL00001619.  Google Scholar

[5]

M. Bourdon and H. Pajot, Quasi-conformal geometry and hyperbolic geometry, in Rigidity in Dynamics and Geometry, Springer, Berlin, 2002, 1-17.  Google Scholar

[6]

G. Cui, Dynamics of rational maps, topology, deformation and bifurcation, Preprint, May, 2002 (early version: Geometrically finite rational maps with given combinatorics, 1997). Google Scholar

[7]

R. L. Devaney, D. Look and D. Uminsky, The escape trichotomy for singularly perturbed rational maps, Indiana Univ. Math. J., 54 (2005), 1621-1634. doi: 10.1512/iumj.2005.54.2615.  Google Scholar

[8]

A. Douady and J. H. Hubbard, On the dynamics of polynomial-like mappings, Ann. Sci. Éc Norm. Sup., 18 (1985), 287-343.  Google Scholar

[9]

M. Gromov, Hyperbolic groups, in Essays in Group Theory, Math. Sci. Res. Inst. Publ., 8, Springer, New York, 1987, 75-263. doi: 10.1007/978-1-4613-9586-7_3.  Google Scholar

[10]

P. Haïssinsky, Géométrie quasiconforme, analyse au bord des espaces métriques hyperboliques et rigidités, Astérisque, 326 (2009), 321-362.  Google Scholar

[11]

P. Haïssinsky and K. Pilgrim, Quasisymmetrically inequivalent hyperbolic Julia sets, Rev. Mat. Iberoam., 28 (2012), 1025-1034. doi: 10.4171/RMI/701.  Google Scholar

[12]

J. Heinonen, Lectures on Analysis on Metric Spaces, Universitext, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4613-0131-8.  Google Scholar

[13]

M. Kapovich and B. Kleiner, Hyperbolic groups with low-dimensional boundary, Ann. Sci. Éc Norm. Sup., 33 (2000), 647-669. doi: 10.1016/S0012-9593(00)01049-1.  Google Scholar

[14]

B. Kleiner, The asymptotic geometry of negatively curved spaces: Uniformization, geometrization and rigidity, in International Congress of Mathematicians, II, Eur. Math. Soc., Zürich, 2006, 743-768.  Google Scholar

[15]

O. Lehto and K. I. Virtanen, Quasiconformal Mappings in the Plane, Springer-Verlag, Berlin, Heidelberg, New York, 1973.  Google Scholar

[16]

C. McMullen, Automorphisms of rational maps, in Holomorphic Functions and Moduli I, Math. Sci. Res. Inst. Publ., 10, Springer, New York, 1988, 31-60. doi: 10.1007/978-1-4613-9602-4_3.  Google Scholar

[17]

J. Milnor, Dynamics in One Complex Variable: Third Edition, Annals of Mathematics Studies, 160, Princeton Univ. Press, Princeton, NJ, 2006.  Google Scholar

[18]

K. Pilgrim and L. Tan, Rational maps with disconnected Julia sets, Astérisque, 261 (2000), 349-383.  Google Scholar

[19]

W. Qiu, X. Wang and Y. Yin, Dynamics of McMullen maps, Adv. Math., 229 (2012), 2525-2577. doi: 10.1016/j.aim.2011.12.026.  Google Scholar

[20]

W. Qiu, F. Yang and Y. Yin, Rational maps whose Julia sets are Cantor circles, Ergod. Th. & Dynam. Sys., 35 (2015), 499-529. doi: 10.1017/etds.2013.53.  Google Scholar

[21]

N. Steinmetz, On the dynamics of the McMullen family $R(z)=z^m+\lambda/z^l$, Conform. Geom. Dyn., 10 (2006), 159-183. doi: 10.1090/S1088-4173-06-00149-4.  Google Scholar

[22]

L. Tan and Y. Yin, Local connectivity of the Julia sets for geometrically finite rational maps, Sci. China Ser. A, 39 (1996), 39-47.  Google Scholar

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